George A. Anastassiou
Parametrized Arctangent Sigmoid Function Based Banach Space Valued Neural Network Approximation
Neural Parallel and Scientific Computations 33 (2025) 25-37

https://doi.org/10.46719/NPSC2025.33.02

ABSTRACT.

Here we present the univariate quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. We perform also the related Banach space valued fractional approximation. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivatives. Our operators are defined by using a density function induced by a parametrized arctangent sigmoid function. The approximations are pointwise
and with respect to the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. We finish with a convergence analysis.

2020 AMS Mathematics Subject Classification: 26A33, 41A17, 41A25, 41A30, 46B25.

Keywords and Phrases: parametrized arctangent sigmoid function, Banach space valued neural network approximation, Banach space valued quasi-interpolation operator, modulus of continuity, Banach space valued Caputo fractional derivative, Banach space valued fractional approximation.